Робота присвячена розвитку нового методу побудови метричної, ймовірнісної та розмірнісної теорій сімейств зображень дійсних чисел на основі дослідження G-відображень (див. [3]). Показано застосування запропонованого методу для розвитку метричної теорії I-Q∞-розкладів дійсних чисел.
The paper is devoted to the development of a new method for the construction of metric, probabilistic and dimensional theories for families of representations of real numbers via of spacial mappings, under which symbols of a given representation are mapped into the same symbols of other representation from the same family, and they preserve the Lebesgue measure and the Hausdorff-Besicovitch dimension (for such mappings the set of points of discontinuity can be everywhere dense). These mappings are said to be G-mappings (G-isomorphisms of representations)[3]. To investigate the DP-properties of such mappings, we study the problem connected with the faithfulness for the calculation the Hausdorff-Besicovitch dimension of fine covering systems which do not coincide with the systems of cylinders - the family of sets which are unions of neighboring cylinders of the same rank, that belongs to the common cylinder of the previous rank. We show application of the proposed method to the development of metric theory of I-Q∞-representations of real numbers.
